The Marvel of the Golden Ratio in Nature

In the Hitchhiker’s Guide to the Galaxy, the supercomputer Deep Thought reveals the answer to ‘life, the universe,and everything’ as 42. Ever since, this simple number has become a punchline all of its own; a wink and a nod to fans of Douglas Adams, and a dismissal of straightforward answers to complex questions. But what if such a number exists? What if a single golden number in nature governs everything from the shape of snail shells to the spirals of galaxies?

There are those who believe exactly that. That one number appears too frequently throughout the natural world to be a coincidence. It’s been called the golden section, the golden number, the golden ratio and even the divine ratio. Mathematicians refer to it as phi or by the Greek letter ϕ.

So, is the idea of a golden number actually true? Is there such a number or golden ratio in nature? If so, what is it and what are the natural occurrences of the golden number? Read on, because we’re doing the maths.

In the simplest terms, the golden ratio, also known as phi, is a fixed number. And that number is approximately 1.618. It’s what’s known as a mathematical constant, which is a fixed number with useful applications and properties. Another example of a constant is 3.142, better known as pi.

Both pi and phi represent relationships between two quantities that frequently interact. For pi, that’s the relationship between a circle’s circumference and its diameter. Phi is approximately 1.618, and is the unique ratio where if you divide a line into two parts, the whole line divided by the longer part is the same as the longer part divided by the shorter part.

Also like pi, phi is an irrational number; that means the digits after the decimal point continue indefinitely and without pattern. So, 1.618 is like an abbreviation. It can also be expressed as 1.61803. Or 1.61803398875. And so on forever.

Part of understanding the divine ratio in nature is its deep connection to the Fibonacci sequence. This is a series of numbers where each number is the sum of the two preceding ones, as in: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Throughout the sequence, the ratio between consecutive numbers closely approximates the golden ratio, becoming closer and closer to it as the sequence progresses.

The golden ratio is often expressed as shapes whose sides fit the ratio, like a rectangle measuring 10cm by 16.18cm. The most famous depiction is that of such a rectangle inlaid with a square whose sides measure 10cm. The space left over after the square is another rectangle which is again inlaid with a square. Each time a square is placed, another rectangle is revealed, creating a spiral of squares of ever decreasing size leading to the bottom right hand side of the original rectangle. This image is usually completed by a spiralling line drawn to follow the decreasing square sizes. This is known as the golden spiral.

For centuries, people have noted the presence of the golden ratio in nature. Indeed, it’s been viewed as a formula for ideal beauty. In the 19th century, psychologist Adolf Zeising claimed to have found numerous natural occurrences of the golden number, including in the patterns of veins in leaves as well as the arrangements of plant and tree branches. So, what other examples of the divine ratio in nature have been claimed?

There have been myriad claims of examples of the divine ratio in nature, including:

Plant Growth Patterns

That the arrangement of leaves around a stem often follows a spiral pattern based on the golden ratio. This allows each leaf to receive maximum sunlight and nutrients, optimising the plant’s growth. However, it’s important to note that while this pattern is common in plants, it’s not universal. The underlying principle is that plants grow new parts where there’s room, resulting in patterns that often, but not always, align with the golden ratio and Fibonacci numbers.

Flower Petals

Flowers provide another stunning display of what’s seen to be the golden ratio in nature. Many flowers have petals in numbers that accord with the Fibonacci sequence, such as lilies with three petals, buttercups with five, and daisies with 21, 34, or 55 petals. This pattern extends to the arrangement of petals, which often exhibit a rotational symmetry corresponding to the golden proportion in nature. The consistent recurrence of this pattern suggests an underlying efficiency and evolutionary advantage in maximising exposure to sunlight and space.

Seed Heads

In plants like sunflowers, the seeds are arranged in spirals that conform to the golden ratio, maximising the number of seeds that can fit in the available space.

Pinecones & Pineapples

These structures exhibit spiral patterns that follow the Fibonacci sequence and approximate the golden ratio.

Tree Branching

The way tree branches form or split often follows a pattern related to the Fibonacci sequence and the golden ratio.

Animal Proportions

The animal kingdom is replete with what are seen as examples of the divine ratio in nature. Some animals, like certain fish species, have body proportions that approximate the golden ratio in the placement of features like eyes, fins, and tails.

Shells & Other Spirals

Spirals are often seen as the most visually striking natural occurrences of the golden number. The nautilus shell is a quintessential example, growing in a logarithmic spiral that maintains the same shape as it increases in size, embodying the Golden Ratio. Similarly, the arms of galaxies and the spirals of hurricanes can follow this divine proportion. Again though,it’s important to note that while these natural phenomena often approximate the Golden Ratio, they don’t always adhere to it perfectly.

Human Anatomy & Beauty

Various proportions in the human body, such as the ratio of forearm to hand length, are said to approximate the golden ratio. However, while some proportions within the human body do come close to the golden ratio, many others do not. In the 1980s, orthodontist Dr. Robert Ricketts published several influential articles supporting this idea. He even argued that the human mind was predisposed towards morphology that accorded with the golden ratio, viewing it as the epitome of “beauty, harmony, and balance.” The essential argument being that facial features more closely aligned to the golden ratio are perceived as more attractive.

The Golden Ratio in Natural Phenomena

Beyond living organisms, the Golden Ratio has been said to manifest in broader natural phenomena, from the way rivers branch to the formation of crystal structures and even the dimensions of hurricanes. The logarithmic spiral, a form consistent with the golden proportion in nature, appears in the formation of these elements, suggesting a fundamental efficiency in the natural processes.

The golden ratio began purely as a mathematical principle, identified by the Ancient Greeks for its common appearance in geometry. They did admire the aesthetics of the golden rectangle, but didn’t call it “golden” or “divine.” Greek mathematician Euclid referred to it as “extreme and mean ratio,” while others called it simply “the section.”

It wasn’t until the 16th century that it began to take on a wider significance. In 1509, a friar and mathematician called Luca Pacioli attributed it with celestial importance in the book The Divine Proportion. The title referred to the golden ratio, only Pacioli decreed it had been sent by heaven directly from the Christian God or, as he puts it, the “Supreme Architect.” The work contained illustrations by none other than Leonardo da Vinci, specifically those that incorporated phi. As for the “golden” moniker, this is usually attributed to mathematician Martin Ohm, who’s thought to have coined it in 1835.

Thus, there have been many proponents of the idea that the golden ratio is an intrinsic part of the natural world. But critics argue that prevalence of the divine ratio in nature is often exaggerated, or the result of confirmation bias or selective measurement. Many of these occurrences are approximations rather than exact matches, not least because the golden ratio cannot be applied precisely because it’s an irrational number. What’s more, measurements are too often imprecise or even subjective, taken between points according to the chosen narrative.

From a psychological perspective, it’s been pointed out that the desire to perceive a golden proportion in nature aligns with people’s propensity to seek patterns even where none exist. This tendency to draw order from the jaws of chaos is called patternicity.

In conclusion, the concept of the golden ratio, or divine ratio in nature is one that has endured for centuries. Whether this is proof of the human need for certainty or a divine plan, there will always be those who seek and can find what can be viewed as natural occurrences of the golden number.